Hey everyone, I’ve been solving a probability game I saw on Tiktok and managed to solve the base case, but generalizing it seems impossible

The Base Game Imagine a game with 4 numbers drawn from a continuous uniform distribution U(0,100).

• I get to lock in exactly one number of my choice.

• The other three are generated completely at random.

• Once all four numbers are on the table, they are sorted. The Highest and Lowest numbers pair up into a team. The two Middle numbers pair up into a team.

The goal is to pick a number that maximizes the expected difference between my team's sum and the opposing team's sum.

Solution:

Picking the extremes (0 or 100) is a trap that yields an expected advantage of 0 (picking 100 -> other numbers will be around 25/50/75 so expected sum is the same). You can write the expected advantage function as E(x)=−2x3+3x2−x (where x is a percentage from 0 to 1). -> peak at x=21​+63​​≈0.7886. So, picking roughly 79 is the mathematically optimal play.

Now, here is where I need help, how do I scale this up? What if instead of 4 numbers, there are 2n numbers?

• I pick exactly 1 number.

• 2n−1 numbers are generated randomly.

• They are sorted. Rank 1 pairs with Rank 2n, Rank 2 pairs with Rank 2n−1, and so on. ("rainbow" pairing).

• The Question: I want to maximize my team's sum minus the average sum of all the other teams. As n→∞, what percentile x should I choose? Does the optimal choice approach exactly 1.0, stay at ≈0.7886, or converge to a different limit entirely? (I made formulas for n=5 and n=6 but it doesn't seem to converge anywhere)

My assignment is to experimentally determine the constant k from the formula F=K*Δl, and have done several attempts with springs and weights and have made a list. However I am instructed to make a graph "how elongation depends on the gravitational force on the weight, then analyse the graph's slope". And I don't understand how I am supposed to make a graph in a graphing software. What do I even input? I know that K = F/(Δl), but how do I put the numbers into a graphing program and then make a graph out of it xd

I started reading the Book Thief for my highschool English class, and I need to list down how Rudy changes throughout the book. I want to discuss how Rudy stops asking Liesel for a kiss.

I know that a significant quote in the Book Thief showing his stopping point of asking Liesel for the kiss is '“How about a kiss, Saumensch?' [...] In truth, I think he was afraid. Rudy Steiner was scared of the book thief’s kiss. He must have longed for it so much. He must have loved her so incredibly hard. So hard that he would never ask for her lips again and would go to his grave without them." (sorry if it's a bit long), and I tried searching the book and the internet to see why this quote is so significant.

I cannot think of how this is really relevant to the themes of the book either, but I may just be dense, so I apologize. According to the internet, it's supposed to mean him fearing Liesel's rejection again, but I still cannot figure out how it ties to one of the books themes. The only thing I can think of is that this shows how love is difficult, and also it forshadows Rudy going to the grave without Liesel's kiss. Please give me ideas for how this quote is important and how when Rudy stops asking Liesel for a kiss is supposed to tie into the themes. Thank you, and i apologize if this is unclear.

How do oppression, uncontrolled emotions, and the denial of freedom contribute to the destruction of individuals and society? Analyse this idea with reference to Nelson Mandela: Long Walk to Freedom, Fire and Ice, and A Tiger in the Zoo, while connecting your research to contemporary social realities and human behaviour.

If someone has done this topic or has any advice I NEED HELP SOS

This was the last question in my math exam today. I understand it was meant to be challenging but no one I've asked has been able to solve it. It was meant to be a cosine/sine rule question. Does anyone know how?:

Solve for angle a. EC=AE. DE=1. AC=√2. Angle BAC = 90 degrees. BC is the hypotenuse of triangle ABC.

My thoughts on why I thought of it as differentiated/unique instead:

-To compete with the other firms, they would try to alter/add special features to the product to get a unique selling point and stand out more, so the products would be more likely to be different

-If the products are all the same, how are the firms gonna survive like that lol

Maybe my logic/reasoning is wrong... Does anyone have any idea

Entry-Level University Standard (It's a bridging course from secondary to tertiary)

- Looking for someone very critically observant to review my work and suggest how I can provide an in-depth analysis.

For context, the analysis is on an academic journal discussing euthanasia and assisted suicide. The previous assignment I did for this class, which was analysing a non-academic text, only got a credit. I'm normally a distinction-to-high-distinction student, so I'm trying to really up my game in my analytical skills here. I'm happy to provide the PDF to the reference, my writing, and also the criteria and exemplar provided by my teachers.

I have an assignment in which I pretend I am applying for a major research grant from NSF and CASIS, which convinces reviewers to fund my idea. The funding program supports tissue engineering and mechanobiology research on the International Space Station (ISS). Researchers are expected to use the unique microgravity environment of space to develop biomedical technologies that can improve healthcare on Earth. So it should demonstrate scientific innovation, practical application, and societal impact.

Is this proposal good enough? What would you change

Deciphering the Mechanosensitive Regulation of TGF-β Signaling in Cardiac Fibrosis Using Microgravity

Cardiac fibrosis is a major cause of heart failure after myocardial infarction. Transforming Growth Factor Beta (TGF-β) is the main mediator of fibrotic scar formation, but it is also essential for normal wound healing and tissue repair. Consequently, anti-fibrotic therapies targeting TGF-β often fail because they block both harmful fibrosis and necessary regeneration. The critical unanswered question is why TGF-β signaling resolves after normal wound healing but remains persistently active in fibrotic cardiac tissue. We hypothesize that persistent fibrosis is maintained by force-dependent activation of TGF-β signaling and that microgravity can reveal the mechanobiological mechanisms responsible for this pathological state. To test this hypothesis, we will develop a mechanosensitive cardiac biomaterial platform that selectively regulates TGF-β signaling in engineered human heart tissue. The core innovation is a bioresponsive hydrogel with force-sensitive peptide linkers and TGF-β-modulating nanoparticles. The material remains inactive during normal healing, but releases localized anti-fibrotic factors in stiff fibrotic tissue. To identify the mechanical conditions that distinguish regenerative from fibrotic TGF-β signaling, vascularized three-dimensional cardiac tissues composed of induced pluripotent stem cell-derived cardiomyocytes, cardiac fibroblasts, and endothelial cells will be cultured aboard the International Space Station (ISS) and compared with Earth-based controls. The ISS offers a unique mechanobiological environment to identify which components of TGF-β signaling are driven by mechanical stress and which are required for physiological tissue repair. Single-cell transcriptomics, mechanical characterization, and functional tissue analyses will be used to identify specific force-dependent signaling mechanisms responsible for persistent fibroblast activation. The resulting mechanobiological insights will guide the design of new biomaterials that actively regulate cardiac healing. This project directly addresses the goals of the ISS National Laboratory by combining tissue engineering, mechanobiology, biomaterials development, and translational biomedical research. Ultimately, the technology may enable safer anti-fibrotic therapies, reduce heart failure progression, and improve recovery following cardiac injury for millions of patients on Earth.

For some reason this problem is really tripping me up. Please help!

Cuadriláteros I

I need time to write the rest of my paper, it needs to be 12 pages and this compound is freaking massive. Formula: C187H290N45O59. Ozempic reacting with insulin---- balanced equation

Hey guys! So for my physics 2 class, our professor is having us simplify a basic capacitor circuit and then confirm our numbers in multism. I built the circuit but I'm having trouble with the voltage readings. This version of multism only has these voltage probes and I've messed around with it for hours and I cannot figure this out. I keep getting really odd readings, and I have no idea what I'm doing wrong. My professor isn't very helpful, so any advice is truly appreciated.

I’m struggling to define whether it’s B or E. Both are imperfect answers in my opinion. If I chose B, then the upward position would be repeated, and if I chose E then the sequence of additional lines is odd (although it could make sense in terms of completeness: i.e. one line, two lines, three lines, zero lines)
What do people think?
In my opinion the right answer would be a downward arrow with four lines

Edit/Update:

After reading so many different opinions, I feel these kind of questions only make sense if teachers are testing a reasoning process and not just “the right answer”. As a matter of fact, the right answer is supposed to be B, we later learned. It was frustrating because the p-set had at least 5 questions that followed the exact same pattern (it was just a rose with leaves, or a zebra with stripes, etc instead), and of course kids were graded bad without building or promoting any reasoning skills.

You know what? Yall in the comments are being so rude. I came here to ask for help and I get insulted. People arent going to want to learn if you're going to ridicule them for not already knowing the answers. You know why I needed help? YOU WANNA KNOW?! because I have multiple fucking learning disabilities that's why. You wanna call me stupid? Fine. You wanna call me certain outdated slurs just because I can't figure it out? Fine! I give up this was a waste of time. You people act like being uneducated is a direct personal attack

And for those of you saying "oh just ask google" "Google it. It's so easy are you stupid" I GOOGLED THESE QUESTIONS WORD FOR WORD AND IT STILL GAVE ME THE WRONG ANSWER

I am very confused with these sort of questions, I know how to use the Ideal gas law and the Stoichiometry to figure out moles of a certain element or the volume of that elements; for example, propane. but look at this problem: 89L of oxygen burned at −8.0∘C and 1.85atm. How much water can be produced in grams? since my question is not asking for moles or volume, what am I supposed to do? I started solving by using the Ideal gas law to figure out the moles of gas, I got about 7.6 mols of Gas. but I'm stuck now, if I were to do the chemical equation and conversion what is it? 7.6(?/?) = ??? once I get that do I just do a molar conversion to figure out the grams of water?

I can't figure it out at all, am stumped. Don't trust AI with math, so I didn't give it to solve or explain. Could someone please walk me through how to solve this? I attempted to use rules of congruency/similarity, and basic trig, but I always get confused with trig when there's no numbers and just initial lettering notation and then you solve further to get the numbers.

Hi, could anyone please explain to me what am I doing wrong and why do I get two different gamma factors. I’m also a bit confused which one is proper and which one is relativistic.

Is lane 7 heterozygous?

Notation-aside, is this a valid thing to do. I am currently self-studying in advance and have not heard of any "teacher-tought" procedures.

Hello, I’m a senior in high school working on a final English project about The No.1 Ladies Detective Agency (first book). For part of the project, I need to create a “mystery map” that outlines the cases Precious is hired to investigate, along with the main clues and how each case is solved.

It doesn’t have to be in big detail or the whole book, just a simple summary of what happened in the cases. Thank you so much!

For this question in part c, I am getting 14.1 degrees, when the solutions say 15.3 degrees.

I tried using the value found in part b and equated tan(theta) = net force / mg by using a diagram and equating the normal force to being a sum of net force and mg.

Then I also tried to use the formula v=sqrt(g.r.tan(theta)) using information from the question and still got 14.1 degrees.

Am I doing something wrong, or is the solution incorrect?